tangherlini’s dissertation and its signiﬁcance for physics...

Tangherlini’s Dissertation and Its

Significance for Physics of the 21th Century

Gregory B. Malykin∗ and Edward G. Malykin†

Abstract: Here we comment on some of the most important re-sults obtained by Frank Robert Tangherlini, in his Dissertation of1958. We show that the main difference between the Tangherlinitransformations and the Lorentz transformations arises from a specialmethod synchronizing two clocks spatially separated in two differentinertial reference frames as was suggested by Tangherlini, which isdifferent from Einstein’s method of synchronization suggested earlier.It is shown, despite the aforementioned circumstance and also despitethe fact that the Tangherlini transformations differ from the Lorentztransformations (in particular, the Tangherlini transformations allowthe velocity of light to be anisotropic in a moving inertial frame ofreference), the Tangherlini transformations provide adequate expla-nations to all known well-verified experimental tests of the SpecialTheory of Relativity. Several possible applications of the Tangher-lini transformations could give an explanation to the effects, alreadypredicted by physicists but not yet registered. In particular, oncethe effects have been experimentally observed (a possible violation ofthe Lorentz-invariance may be involved), the effects might be moreproperly described with use of the formalism of the Tangherlini trans-formations.

During the last seven decades, physicists have discussed kinematic the-ories which are claimed as alternatives to the Special Theory of Relativ-ity, or are based on transformations of the spatial coordinates and timefrom one inertial frame into another one which differ from the Lorentztransformations [1]. Meanwhile, despite the fact that several suggestedtransformations can explain numerous basic experiments of the Spe-cial Theory of Relativity, in particular — the Michelson-Morley exper-iment [2, 3], not one of the suggestions except the Tangherlini trans-formations [4] and the Sjodin transformations [5], which generalize theformer, are able to give a proper explanation to all known experimentaltests of the Special Theory of Relativity, in particular — the interfer-ence experiments, the measurements of the transverse Doppler effect,and the increased lifetimes of decaying high energy particles. In addi-tion, these alternative transformations can deal with the effects in therotating frame of the Sagnac experiment [6–10] in which the Einstein

∗† Institute of Applied Physics, Russian Academy of Sciences, Ulianova Str. 46,Nizhni Novgorod 603950, Russia. E-mail: [email protected] (G. B.Ma-lykin). E-mail: [email protected] (E.G.Malykin).

122 The Abraham Zelmanov Journal — Vol. 2, 2009

synchronization procedure cannot be used nor that involving slowly-moved clocks, as was pointed out for the latter case by Tangherlini ina thought experiment described in his general relativity lectures [11],and verified some years later in the well-known Hafele-Keating experi-ment [12, 13].

It should be noted that, in the framework of most of the aforemen-tioned transformations, due to the specific methods of synchronizationof spatially separated clocks, the velocity of light is isotropic only ina single (preferred) inertial reference frame. In the framework of thesetransformations, the velocity of light still remains isotropic and invari-antly constant (equal to c), and independent of the velocity of the source,in agreement with the Special Theory of Relativity, while, in contrast,the value of the transverse Doppler effect predicted according to thesetransformations (excluding those of Tangherlini and Sjodin) differs fromthe value predicted by the Special Theory of Relativity, and hence arein disagreement with present experimental evidence. Recall that theconstancy of the velocity of light in vacuum in any direction and itsindependence from the velocity of the radiating source, in an arbitraryinertial frame of reference, are well-verified experimental postulates ofthe Special Theory of Relativity, which follow directly from the Lorentztransformations [14].

Not one of the transformations alternative to the Lorentz transfor-mations has been so actively discussed in the scientific press as theTangherlini transformations obtained in 1958 [4]. The discussion hasled to a large amount of literature on the subject. The interest to theTangherlini transformations has been due to the fact that they can beuseful in the search for a theoretical explanation of a possible “delicateviolation” of the Lorentz-invariance (this is employed sometimes in orderto explain several exotic phenomena such as the origin of high-energycosmic beams, the origin of dark matter and dark energy, several cos-mological models, and also quantum gravity models, see [15–24]). Anti-relativists suggest that the Tangherlini transformations can be used asa proof for an absolute (preferred) inertial frame of reference, which isconnected with the “luminiferous ether”, and as a proof for the viola-tion of Lorentz invariance in physical phenomena. Hence, they declarethat the Tangherlini transformations refute the validity of the SpecialTheory of Relativity. Other researchers emphasize that the Tangherlinitransformations are not merely an alternative but an equal replacementfor the Lorentz transformations, hence the Special Theory of Relativ-ity is no more than one of many equivalent theories describing physicalprocesses from the viewpoints of observers located in two different iner-

Gregory B. Malykin and Edward G. Malykin 123

tial frames. Probably, not many of the participants who have discussedthe Tangherlini transformations (this discussion started in 1977, afterReza Mansouri and Roman U. Sexl published the paper [25]) have readTangherlini’s original Dissertation of 1958 [4], wherein he deduced thetransformations and studied their applications to classical theory andquantum theory. Most scientists know only the later publications of1961 and 1994 [11, 26], which give the transformations with differentdeductions and fewer applications. For more than fifty years the Dis-sertation was able to be accessed, as a manuscript copy, only at theStanford University Library. Its appearance here in the present issueof The Abraham Zelmanov Journal [4] marks its first date of publica-tion. Moreover, it is prefaced by the very interesting discussion writtenby Frank Robert Tangherlini himself [27], and is accompanied by hisfine comment [28] to §8.1 of [11], where he gives an explanation aboutseveral properties of the Maxwell electrodynamics on the basis of thetransformations.

In his Dissertation [4], Tangherlini uses a special system of unitswhich is specific to several studies on the theory of relativity. In thissystem, time has the same dimension as length (so that Tangherlinimeans time t as a regular time multiplied by c, the velocity of light invacuum), while velocities are dimensionless. In other words, a velocityhe uses is a regular velocity divided by c. As a result, the velocityof light in vacuum in this system of units equals 1. This makes theunderstanding of the obtained result a little complicated. Meanwhile,in the preparation of the manuscript for publication, the Editor of thejournal and the Author have decided to keep the original notations andthe system of units unchanged for historical reasons. In contrast, inwhat follows, we will use the regular system of physical units whendiscussing the Tangherlini transformations and their sequels in physics,which will make their understanding a lot easier.

The main task we are targeting in this paper, which accompaniesTangherlini’s Dissertation [4], are comments on the results obtainedtherein. The biography of Tangherlini and the circumstances of his Dis-sertation were given in detail in the publications [29, 30]. We thereforelimit ourselves to only a brief survey of the events.

Frank Robert Tangherlini was born on March 14, 1924, in Boston(Massachusetts, USA), in the family of a worker. In 1943, during theWorld War II, he volunteered to be drafted into the U.S. Army, and latervolunteered, after arriving in England in 1944 as an infantry replace-ment, to serve in the 101st Airborne Division as a paratrooper. Afterparachute training in England, he was dispatched to France, and then


fought in Belgium and Germany. He also participated in bloody Battleof the Bulge in the Ardennes, where many of his paratrooper friendswere killed in action. In 1946, he returned to the USA and was honor-ably discharged from the U.S. Army. In 1948, he obtained his BSc (cumlaude) from Harvard University, followed by an MSc from the Universityof Chicago (1952), and a PhD from Stanford University (1959). During1952–1955 he worked in the space industry in San Diego, and during1958–1960 he served as a National Science Foundation postoctoral fel-low, and from 1960–1994, he worked as a faculty member, or on the staffof different universities and research institutes in the USA and Europe.At the present time he is retired. He lives in San Diego, California,where he is still active in scientific research. The field of his scientificinterest is very wide and covers the Special Theory of Relativity, di-mensionality of space, relativistic cosmology, Mach’s principle, etc. Hismost important scientific results are the transformations he deducedwhile working on his Dissertation of 1958. Later, these became widelyknown as the Tangherlini transformations (in 1958 Tangherlini used an-other, less successful term the Absolute Lorentz Transformations). Hischief supervisor in this work was Sydney David Drell (b. 1926), the well-known expert in Quantum Electrodynamics who later became a friendof Andrew D. Sakharov. At the initial stage of Tangherlini’s work, hisactual supervisor was Donald R. Yennie, also well-known for his workon quantum electrodynamics, and who had been appointed his super-visor by Leonard Isaac Schiff (1915–1971) who was the chairman of theStanford physics department, and was also responsible for Tangherlinireceiving a fellowship to continue his graduate studies at Stanford. InJune, 1958, Tangherlini reported his results at a colloquium of physics atStanford University, and then submitted the Dissertation to the PhysicsSection of the Graduate Division. Positive reviews of the dissertationwere submitted by Drell and Schiff. Afterwards, following the receipt ofa National Science Foundation postdoctoral fellowship, Tangherlini wentto the University Institute for Theoretical Physics in Copenhagen (thisInstitute was later called Bohr Institute). While abroad, Tangherliniwas graduated with a PhD degree from Stanford University in absentia.

The direct and inverse Tangherlini transformations are introducedby means of a special method of synchronization of spatially separatedclocks located in two inertial frames of reference, one of which is takento be the preferred frame in which the one-way speed of light is c, andthe other frame is moving relative to it with speed v, in which thespeed of light varies with direction and v as discussed below. In thismethod, the clocks are synchronized by signals travelling with an in-


finitely high speed. Already in 1898, Henri Poincare in his paper [31]and his address [32] delivered on September 24, 1904, at the Congressof Arts and Sciences in St. Louis (Missouri, USA), pointed out the sig-nificance of faster-than-light synchronization methods in effecting theoutcome of the measurements of the velocity of light, and some yearsearlier, in 1898 [31] had given an interesting discussion of the meaningof simultaneity. This problem was also considered in 1934 by LeonidI. Mandelshtam (1879–1944), in his lectures on the physical grounds ofthe theory of relativity [33] read in 1933–1934 at Moscow University.Sergey M. Rytov (1908–1996) restored Mandelshtam’s lectures after hisdeath, on the basis of the records made by Gabriel S. Gorelik, Maxim A.Divilkovski, Michael A. Leontovich, Z. G. Libin, who heard the lectures,and also on the basis of Mandelshtam’s draft notes. Then Rytov pub-lished the lectures in 1950 [33] (second edition [34] was printed in 1972).In his lecture, held on March 10, 1934, Mandelshtam was engaged inpolemics with anti-relativists on the formulation of the causality prin-ciple in the Special Theory of Relativity, and on the problem of thesimultaneity of events in different inertial frames of reference. In thesepolemics, Mandelshtam focused the attention of the listeners on thefact that Reichenbach’s method of synchronization does not violate thecausality principle∗. He said:

“Thus, the requirement that the causality principle must not beviolated in the definition of simultaneity can be simply satisfied. . . if there would be a signal travelling at infinite speed, the re-quirement that the causality principle must be true would give asimple condition universally to all frames [of reference]. . . . Thus,it is required to understand that there should not be such a faster-than-light signal, which may generate action. . . . If I make use ofa process, which cannot produce action, this does would violatethe causality principle. . . . Many researchers tried to give sucha definition to simultaneity, which they believed does not dependon the possibility of its empirical determination, but rather arisesfrom the supposition that there is an apriori simultaneity”.†

Proceeding further, Mandelshtam considered a possibility of synchro-nization of spatially separated clocks, in different inertial frames of ref-

∗Here Mandelshtam obviously considered the method of synchronization of spa-tially separated clocks suggested by Hans Reichenbach (1891–1953) in [35,36], whichis quite different from the synchronization method suggested by Einstein [14].

†Here Mandelshtam attempted to focus the attention of the listeners on the factthat synchronization of clocks by infinite speed signals leads to simultaneity in allinertial frames of reference.


erence, by phase velocity signals (phase velocity may be as faster thanlight as possible). Such signals transfer neither action nor information.Therefore, if a phase velocity approaches infinity, the causality princi-ple is not violated. Unfortunately, Mandelshtam, in his lectures [33,34],arrived at the conclusion that this method of synchronization is unableto be realized in practice. This is because he considered the phase ve-locity of a signal produced by a mechanical device like scissors, namely— the motion of the cross-point of a scissors, which is realized withonly a finite phase velocity. Therefore Mandelshtam had not realizedthe principal step in this research: he had not deduced transformationof the spatial coordinates and time from one inertial reference frame toanother one, synchronized by infinite speed signals. It is probable that1934 was too early a time for this principal step that was made only 24years later, by Tangherlini.

In this connexion, we should emphasize an interesting and impor-tant paper, published by Albert Eagle, the British mathematician who,already in 1938, was extremely close to the Tangherlini transformations.In his paper [37], Eagle considered a method of synchronization of spa-tially separated clocks by a mechanical shaft, rotating by a clock enginelocated at its centre. The clocks under synchronization were fixed atthe shaft’s butt-ends. This method of synchronization was also con-sidered in Eagle’s second paper [38]. Albert Eagle, being an obviousanti-relativist∗, held the erroneous belief that a mechanical shaft ro-tates as a perfectly rigid body, so that torsional perturbations wouldtravel instantaneously along it. Meanwhile, as we know, the perturba-tions do not propagate instantaneously along the shaft, but with thesound velocity specific to the substance of the shaft. This velocity ismany orders slower than light†. Eagle targeted his publications [37, 38]as a proof for the reality of an absolute (preferred) reference frame con-

∗This fact concerning his personality can be easily concluded from his papers[37, 38] and, especially, from his other paper [39].

†We note that, aside for this simplest reason, synchronization of clocks by arotating shaft is sensitive to the relativistic change of the figure of the shaft in aresting inertial frame of reference, as was considered by Ives (1882–1953), in [44],in the example of a rotating double Fizeau cogwheel (actually, a double obturator).When Stefan Marinov (1931–1997) performed his single-way measurements of thevelocity of light using two mechanically connected systems consisting of rotating mir-rors [45], he met a criticism from the side of Simon James Prokhovnik (1920–1994)who pointed out the inconsistency of this method of synchronization [46, 47]. De-spite this criticism, Marinov continued measurements based on this synchronizationmethod, but with another mechanical system which was similar to the previous one(it was a modified double obturator called by him the “coupled shutters” system).See [48] and literature referred therein, for Marinov’s experiments.


nected with “luminiferous ether”. Meanwhile, in spite of his incorrectconsiderations on the basis of the Newtonian views on absolute simul-taneity which were already obsolete in 1938, Eagle [37] arrived at thetransformations of the spatial coordinate x and time t associated withan observer’s inertial frame of reference to the corresponding space-time coordinates in another inertial reference frame in the x-direction.Eagle’s expressions for the transformation of x and t agree with the cor-responding expressions of the Tangherlini transformations (1). Directand inverse transformations for the transverse coordinates y and z werenot discussed by Eagle, but he did note that the inverse transforma-tion for the x and t coordinates was not of the same form as the directtransformation, as did Tangherlini.

Unfortunately, Eagle’s key paper [37] was ignored by the scientificcommunity from the time commencing in 1938 when it was first pub-lished. The sole reference to this paper, which we have found in thescientific literature, appeared in Eagle’s second paper [38]∗. Why? Nosimple answer can be given to this question. In any case, whateverbe the answer, the real physical sense of the transformations was onlyachieved 20 years later in Tangherlini’s thesis of 1958, in which theirderivation and application is discussed clearly and in detail. DmitriRabounski, who also discussed Eagle’s papers [37, 38], commented thissituation as follows [41]:

“After reading Eagle’s paper of 1938, and his following paper, Iarrived at the conclusion that Eagle obtained his transformationsof the spatial coordinate and time, which particularly meet theTangherlini transformations, as a result of a formal blindfold ofcombinations, not a systematical research. Besides, he was mis-taken about the obtained result due to his erroneous disbelief inthe theory of relativity. His appeal is that the presence of a physi-cal medium fixed to (accompanying) the space as a whole is in con-tradiction with the theory of relativity. This is absolutely wrongand seems naive. He merely had no clear understanding of thetheory of relativity — the geometrical theory of space-time andmatter — and how the theory works. The second paper authoredby Eagle is a logical continuation of his erroneous views on thetheory of relativity, based on the principles of classical physics.According to him, “true synchronization” is synchronization of

∗Max Jammer in his book Concepts of Simultaneity [40], published in 2006, refersto authors who criticized Eagle’s synchronization procedure. Tangherlini refers toJammer’s book in his Preface of 2009 to “The Velocity of Light in Uniformly Moving

Frames” [27].


clocks in the Newtonian sense, while all the remaining methods ofsynchronization leads to non-observable (imaginary) effects, whichare unable to be registered in real measurements. In particular,Eagle referred to his first paper of 1938 as an example of howthe use of “true synchronization” manifests that all effects of thetheory of relativity are non-real, imaginary. Eagle’s conclusion isin contradiction to the many decades of experimental verificationsof the theory of relativity performed in different experiments withhigh precision of measurement. In fact, Eagle had no idea aboutthe real physical sense of his formal mathematical deduction. Hewas in captivity of the views of classical physics, and failed thepossibility of all other research methods in physics (the theoryof relativity, for instance). This is the same as, given all expla-nations on the basis of the wave theory of light, denying all theresults obtained in the framework of the corpuscular theory”.

In continuation of this discussion, Tangherlini wrote recently in his pri-vate letter dated June 01, 2009 [42]:

“With respect to Dr. Rabounski’s penetrating criticism, I wouldadd further that Eagle’s obvious anti-relativity bias led him toreject the general theory of relativity, and this was most ironic,and indeed tragic, because a key principle in general relativityis Einstein’s principle of general covariance which permits arbi-trary transformations of the coordinates, and hence permits thetransformation Eagle was using when supplemented by the trans-formation for the transverse y and z coordinates. In fact I wouldlike to take this opportunity to emphasize to you and Dr. Raboun-ski that I probably would never have undertaken the writing of mythesis were it not for the logical justification provided by general

covariance for making use of such a transformation when supple-mented by the metric postulate as mentioned in the Introductionto my thesis”.

Meanwhile, we should pay tribute to Eagle’s paper [37], despite the factthat Eagle himself misunderstood the physical sense and meaning of hispioneering result. He was the first person who obtained, in a purelyformal way, a part of the common transformations of the spatial coor-dinates and time which were developed in detail later by Tangherlini,and are known as the Tangherlini transformations.

What is interesting is that, in already 1922, a method of synchroniza-tion of spatially separated clocks by means analogous to a locomotivewheel pair was suggested by Carl Axel Fredrik Benedicks, the Swedish


physicist who considered a rotating wheel (that can be any space body,the Earth for instance) as an original time-giver [43]:

“Let us return to the top, which seems to offer a somewhat clearerexample of an original time-giver. . . . the question arises, how itstime-indication may be applied to another process which mightoccur at a great distance. The simplest example is that wherethe second process is also an identical, rotary one that is, anotheridentical top, rotating on the same fixed plane. Simultaneity orsynchronism is said to prevail if a radius vector of the one is alwaysparallel to a corresponding radius of the other. We ask, in whatway can this definition be applied? Evidently, it can be realizedin the way used to synchronize two paired wheels of a locomotive;that is, a solid movable connecting rod is pivoted at the end pointsof two radii, where the length of the rod is equal to the distanceseparating the two axles. As the two radii have been assumedto be equal, they will during the motion also remain parallel. Inprinciple this will fully define simultaneity, so long as the axes ofrotation remain parallel. The first rotating body, A, is the stan-dard which determines time; the second body, B, may act as aclock or timepiece, by exactly reproducing A’s time. . . . We saythat two distant clocks are synchronous, provided that their hands

are moving as though their axles were connected by one rigid axle,

consisting of an absolutely solid body. This is the simpler form ofsynchronizing frequently used, for example, in synchronizing twowagon-wheels belonging to the same axle. . . . This definition ofsynchronism is precise, and has no ambiguity. It is founded onlyupon the fundamental basis for all measurement of time the ac-cepted unchangeability of the rotation process chosen as standardand upon pure geometry the fundamental basis of which is theexistence of the absolutely solid body”.

This method of synchronization meets that suggested by Eagle [37,38].However, in contrast to Eagle, Benedicks [43] had no idea about respect-ive transformations for the spatial coordinates and time. Also, Bene-dicks assumed that there are absolutely rigid bodies, i.e. bodies in whichsignals propagate instantaneously from one end to the other, and hencehis proposal encounters the same difficulty as Eagle’s proposal, and aswas remarked earlier, he seemed unaware that torsional waves, or wavesin material bodies more generally, propagate with a finite velocity.

Tangherlini also suggested another method of synchronization of spa-tially separated clocks, the “external synchronization”, which was given


in detail later, in his general relativity article [11]. He writes [42]:

“It was the difficulty I encountered with the failure to find em-pirical evidence to support faster-than-light signalling that led meto turn to external synchronization, which represents an experi-mental procedure one can carry out now with existing equipment,that can be used to verify the basic predictions of the transforma-tion empirically, as you discuss. . . . Actually I did not make any

hypotheses as to how the instantaneous signals would arise in mythesis, so my synchronization would include the light spot method,but would be more general. The tachyons (as they are now called)were only mentioned in the concluding Chapter 12 of the thesis asa conceivable way of implementing the instantaneous synchroniza-tion, but the argument in the body of the thesis is independent

of any way of achieving such a synchronization; in a sense, there-fore, it is purely mathematical of the type: if we assume X , thenthe following is the case. . . . Actually, I didn’t mention externalsynchronization there, until my later publication An Introduction

to the General Theory of Relativity (see in Supplemento al Nuovo

Cimento, 1961, ser.X, vol. 20, 1–86)”.

Many physicists believed (and, indeed, still believe) this to be impos-sible, because superluminal speeds are attributed only to hypotheticalfaster-than-light particles — tachyons∗ [49,50]. In fact, synchronizationof this kind can be performed. The easiest case is the one where allclocks of both the moving inertial frame of reference and the restingframe of reference are located along the same line. To perform such

∗Tachyons — faster-than-light particles were first coined in the scientific pub-lications on the theory of relativity in the pioneering paper of 1962 [51], authoredby Olexa-Myron Bilaniuk, Vijay Deshpande, and George Sudarshan, who worked inDepartment of Physics and Astronomy, University of Rochester, New York. Theypointed out the historical fact that, in pre-relativity times, Thomson, Heaviside,and Sommerfeld had considered particles moving faster than the velocity of light invacuum. They considered the possibility of such particles in the framework of theSpecial Theory of Relativity. This term, “tachyon”, was introduced into science byGerald Feinberg (1933–1992) five years later, in 1967 [52], while Feinberg worked atRockefeller University, New York, the same city as his predecessors. In this back-ground story, many researchers and historians of science missed the fact that FrankRobert Tangherlini was actually the first person who considered the possibility oftachyons and faster-than-light signals, in a very general sense, in the framework ofthe Special Theory of Relativity, already in 1958. Unfortunately, most papers andbooks on the history of tachyons do not mention this fact. See [53], for instance.Meanwhile, the most important surveys of this theme such as [50, 54] referred toTangherlini’s goal with respect to this problem. In the last decade [55, 56], thepossibility that tachyons had been produced was investigated at CERN.


“instantaneous” synchronization in this case, the “light spot method”should be used, where a light spot travels with a phase velocity whichmay exceed the velocity of light. This method was suggested by Vi-taly L. Ginzburg (b. 1916) in [57], and, in more detail, in the commonpaper by Boris M. Bolotovskiı (b. 1928) and Vitaly L. Ginzburg [58].In these papers, the motion of a light spot along a screen was con-sidered, where the light spot was due to a light beam produced by asource (searchlight) rotating with an angular velocity Ω. If two points,say A and B, are equally distanced at a very large distance R fromthe searchlight, the linear velocity v of the light spot on the screenshould satisfy the condition v=RΩ≫ c. Of course, a light spot can-not transfer energy/information from A to B (with any velocity, bothsubluminal and superluminal): photons coming in A never come to B,hence the causality principle is still true, without violation in the exper-iment. Thus, huge speeds much faster than light are attributed to thelight spots produced by the radiation of pulsars [57–60]. More detailsabout synchronization of distant clocks by the light spot method areconsidered in our works [60–62].

Thus, Frank Robert Tangherlini has deduced transformations forthe spatial coordinates and time from one inertial frame of reference toanother one in the case where clocks located in both inertial referenceframes are synchronized by infinite speed signals in the sense of thatwhich was considered above.

Although not discussed in his Dissertation [4], in his later inves-tigations, Tangherlini also considered another method of synchroniza-tion not involving faster-than-light signals, as described in detail in Ap-pendix A of his 1994 paper Light Travel Times Around a Closed Uni-

verse [26]. This is the so-called “external synchronization”, consisting oftwo steps. First, light signals synchronize clocks which are spatially sep-arated from each other in the same resting (“preferred”) inertial frameof reference. Then, these already synchronized clocks of the “preferred”inertial frame are used for synchronization of clocks of moving inertialframes of reference during those moments of time when each of the mov-ing clocks physically meets one of the resting (synchronized) clocks inspace. In this method of synchronization, inertial frames of referenceare non-equal to each other: the inertial frame where clocks were firstsynchronized is “preferred” to all remaining (moving) inertial frames.

What is interesting is that Mandelshtam, already in 1934, considereda method of synchronization, which would also give rise to absolutesimultaneity in the inertial reference frames K and K ′, as he describedin his lectures [33, 34]:


“. . . Suppose we have a frame [of reference], and one set up syn-chronization in it by a method, for instance — by the Einsteinianmethod . . . Let us also have another frame [of reference]. I couldarbitrarily set up synchronization in this other frame [of reference]so that clocks located in it would always show the same time asthat displayed by the clocks of the first frame [of reference]”.

This is however not the same as Tangherlini’s method of external syn-chronization. Tangherlini, when found Mandelshtam’s achievements inthe 2000’s (his lectures were published only in Russian, so inaccessibleto most scientific community), says [42]:

“This is not true for my transformation because (think of theclocks on the train going past the station) after the brief momentof synchronization, the clocks on the train run more slowly thanthe clocks on the station platform, this is why high energy muonsdecay significantly more slowly than muons at rest in the labora-tory!”

Meanwhile, with use of both methods of synchronization, not all in-ertial frames of reference are equal: that inertial frame of reference,wherein the first synchronization was performed, becomes preferred toall remaining inertial reference frames. In particular, Mandelshtamwrote [33, 34]:

“. . . in this case, we cannot require the relativity principle. . . .When Einstein says that the relativity principle takes a place innature, this means that, if all definitions of age are given equallyin any frames [of reference], events [in any reference frames] willbe processed equally”.

Mandelshtam had not realized any step towards respective transforma-tions in this case. This fact manifests, again, the outstanding thinkingand courage Tangherlini showed in scientific research: he worked with-out looking back on the authoritative persons in science, mostly conser-vators, so he reached the advanced results which were out of the accessfor many other scientists.

Consider the direct and inverse Tangherlini transformations

x′ = γ (x− vt) , x = γ−1x′ + γvt′,

y′ = y, y = y′,

z′ = z, z = z′,

t′ = γ−1 t , t = γ t′ ,

(1)


where x, y, z, t and x′, y′, z′, t′ are the spatial coordinates and time inthe inertial reference frames K and K ′ respectively, c is the velocity oflight in vacuum, v is the velocity of the reference frame K ′ with respectto the preferred reference frame K (we assume it to be moving along

the x-axis), while γ=1/√

1−v2/c2 is the so-called Lorentz-factor.Comparatively, the classical Lorentz transformations are

x′ = γ (x− vt) , x = γ (x′ + vt′) ,

y′ = y , y = y′,

z′ = z , z = z′,

t′ = γ (t− vx/c2) , t = γ (t′ + vx′/c2) .

(2)

It is obvious that, according to the Tangherlini transformations (1),time t′ of a moving inertial frame of reference is slower than time tby the factor γ. That is, the Tangherlini transformations lead to thetransverse (relativistic) Doppler effect as in Einstein’s Special Theory ofRelativity [14]. The direct Tangherlini transformations, which are thefirst column in formula (1), differ from the direct Lorentz transforma-tions, the first column in (2), by only the transformation of time due tothe different methods used in the synchronization of clocks in inertialframes of reference.

Both direct and inverse Lorentz transformations take the same formupon reflecting the velocity, while the direct and inverse Tangherlinitransformations are not (see the final part of Tangherlini’s comment [28]for detail). Besides, in contrast to the Lorentz transformations, whichform a Lie group whose parameter is the velocity, the Tangherlini trans-formations do not form such a group, since as is clear from (1), theinverse of the Tangherlini transformation does not have the same formas the direct transformation, nor do the product of two such transfor-mations have the same form as the direct transformation. On the otherhand, Tangherlini [63] has pointed out that the transformations aremembers of the group of linear space-time transformations that keepsimultaneity invariant, of which the Galilean transformations form aproper Lie subgroup. Also, along with the Lorentz and Galilean trans-formations, the Tangherlini transformations are also members of thelinear unimodular group, i.e., the group of linear space-time transforma-tions with unit determinant, as mentioned in his thesis. The asymmetryof the direct and inverse Tangherlini transformations is connected withthe fact that two inertial reference frames K and K ′ having Galilean(rectangular Cartesian) coordinate frames are equal in the framework of


the Lorentz transformations, while in the framework of the Tangherlinitransformations the inertial reference frames are non-equal: K still pos-sesses the Galilean (rectangular) coordinate frame because the observeris resting in this frame, while the coordinate frame of K ′ is non-Galilean(oblique-angled).

Here we should note that the term “Lorentz transformations” wasintroduced by Henri Poincare [64, 65]∗ in 1905, and he also discussedtheir group properties, as did Einstein in that same year.

Proceeding from the Tangherlini transformations (1), one can obtaina formula connecting the co-linear velocities V and V ′, measured in theinertial reference frames K and K ′ respectively, or, similarly, the law ofcomposition of velocities according to the Tangherlini transformations

V ′ =V − v

1− v2

c2

(3)

as originally obtained by himself in [4]. As visible, this formula is quitedifferent from the law of composition of velocities V ′ = V − v

1− vV/c2, which

holds according to the Lorentz transformations.Having the law (3) as a base, Tangherlini [4] has also obtained a

formula for the velocity of light in vacuum, measured in a moving inertialreference frame K ′, and referred to as c ′

c ′ =c

1 + vc cos θ

′, (4)

where the angle θ′ is counted from the x′-axis in the moving inertialframe K ′.

In a common case, where light travels in an optical medium whoserefraction coefficient is n, measured in a resting inertial reference frameK, the velocity of light in a moving inertial reference frame K ′, accord-ing to Tangherlini [4] takes the form

c ′ =c

n+ vc cos θ

′. (5)

It is clear that formula (4) explains the results obtained in theMichelson-Morley experiment [2, 3] and the Kennedy-Thorndike exper-iment [67]. This is because, following from (4), the total time of light’stravel forward and backward does not depend on the velocity v of the

∗Hendrik Antoon Lorentz (1853–1928) passed through a long way, full of manytests, to his understanding of the Special Theory of Relativity. He stopped hisresearch when was at a minor step from the acquisition of the transformations [66].


inertial reference frame K ′ moving with respect to the preferred iner-tial reference frame K. Moreover, as was shown in [1], the Tangherlinitransformations provide a clear explanation to all interference experi-ments targeted on checking the Special Theory of Relativity, in partic-ular — the Sagnac experiments [6–10]. And also, as remarked above(see page 121), Tangherlini synchronization can be used to carry outcalculations in a rotating frame of reference in which it is not possibleto synchronize clocks by the standard methods of Special Relativity.

A simple explanation of the physical sense of the Tangherlini trans-formation is given by Giancarlo Cavalleri and Carlo Bernasconi [15].The invariance of the velocity of light and the non-conservation of thesimultaneity of events, spatially separated in different inertial framesof reference, are often considered as specific properties of the SpecialTheory of Relativity. Therefore, Tangherlini [4] formulates his ownversion of the theory of relativity, where the absolute simultaneity ofspatially separated events is allowed, and the velocity of light becomesnon-invariant. In the framework of the “standard” Special Theory ofRelativity, the velocity of light determined by formula (4) should not bea physical (observed) velocity, but a coordinate velocity. Actually, theconservation/non-conservation of simultaneity and the invariance/non-invariance of the velocity of light depend on the employed method ofsynchronization of clocks, located in different inertial frames of refer-ence. This fact leads to an infinite number of versions of transfor-mations from one inertial reference frame to another one [5]. In thisrow, two kinds of transformations — the Lorentz transformations andthe Tangherlini transformations — are the limiting cases of the Sjodintransformations [5].

Michael A. Miller, Yuri M. Sorokin, and Nikolai S. Stepanov [68],and then Anatoly Logunov [69] take under consideration an arbitrarylinear transformation from Galilean coordinates x, y, z, t of an inertialreference frame to coordinates X , Y , Z, T of a so-called “generalized”inertial reference frame. Such transformations mean non-orthogonal(oblique-angled) coordinate netsX , Y , Z, T [68] described by the metrictensor whose components are constants. As was shown in [69], if weassume different parameters of the components of the metric tensor inthe “generalized” inertial frame of reference, different formulae can beobtained for the components of the coordinate velocity of light (VX ,VY , VZ) in the “generalized” frame, and the velocity is anisotropic ina general case, while in contrast, in the Special Theory of Relativity,as is well-known, the speed of light takes the same value for all inertialframes of reference and is isotropic.


Tangherlini [70] was able to show that when the standard canon-ical commutation relations of Quantum Mechanics in the Schrodingerrepresentation are enlarged to included the energy and time, and one as-sumes that the energy and momentum transform as a four-vector, thesecommutation relations are not only invariant under the Lorentz transfor-mations, but under all non-singular linear space-time transformations,which would include the general transformations discussed above.

The main advantage of the Lorentz transformations, in contrast tothe other kinds of transformations of the spatial coordinates and time,consists in the Einsteinian method of synchronization of spatially sep-arated clocks that keeps the velocity of light isotropic and constant intransferring from one inertial frame of reference to another one. In “gen-eralized” inertial frames of reference, which are a result of the Tangher-lini transformations in particular, no so-called pseudo-forces (the cen-trifugal force of inertia, or Coriolis’ force, for instance) appear as innon-inertial frames of reference where such forces play the role equal tothe force of gravity. In this connexion, incorrect claims, from the pastto the present, can be found in the scientific publications. Thus, HansC. Ohanian [71] claimed, incorrectly, that the Reichenbach method ofsynchronization of clocks [35, 36], upon being realized in an inertialframe of reference, should inevitably lead to the formal appearance ofthe pseudo-forces in the inertial reference frame.

At first, the Tangherlini transformations did not attract much ofthe attention of scientists. This situation changed after 1977, when ananisotropy in the Cosmic Microwave Background Radiation had beendefinitely verified in observations on board a U2 sub-stratosphere air-plane performed by George Smoot’s team [72]∗. In fact, this means thatthe inertial frame of reference connected with the Earth moves in thecosmos with a velocity of about 360km/sec with respect to a preferredinertial frame of reference, in which the Microwave Background Radi-ation is “most” isotropic and the common momentum of all masses ofthe Universe is probably zero. As a result of the experimental success,different suggestions arose to the origin of the observed anisotropy in theCosmic Microwave BackgroundRadiation as due to the anisotropy of thevelocity of light, so the Tangherlini transformations became of interest.The first persons who turned our attention to the Tangherlini transfor-

∗The dipole-like anisotropy was first observed in the ground-based observationsperformed by Edward K. Conklin in 1969 [73], then studied in the balloon observa-tions by Paul S. Henry in 1971 [74] and by Brian E. Corey and David T. Wilkinsonin 1976 [75]. The main reason for Smoot’s success of 1977 [72], and his fame whichfollowed later, was very certain observations of the anisotropy.


mations as a possibility of explaining the results of the Michelson-Morleyexperiment [2, 3], following Tangherlini himself∗, were Reza Mansouriand Roman U. Sexl [25]: they said in the bibliography to their firstpaper that the transformation had been considered by Tangherlini. Af-terwards, many papers were published, wherein the Tangherlini trans-formations were employed: see, for instance, [5, 15, 71, 76–91].

There are also numerous papers wherein the Tangherlini transfor-mations were “re-discovered” anew. These are Stefan Marinov’s publi-cations of the 1970’s [92–94], the paper of 1992 [95] authored by ErnestW. Silvertooth and Cynthia K. Whitney, the papers [96–98] publishedby Nikolai V. Kupryaev commencing in 1999, and the paper of 2001 [99]by Juri A. Obukhov and Igor I. Zakharchenko. Looking along the sci-entific literature, we found a note on the absence of priority concerningthe earliest of the “re-discovering” papers: Giancarlo Cavalleri and Gi-ancarlo Spinelli [100] commenting on the transformations appearing inMarinov’s publications of the 1970’s, and claimed by him as his ownoriginal achievement, gave the priority to Tangherlini who had actu-ally obtained these already in 1958, although they were not publishedin a journal until 1961 [11], and it was to this article to which Sexland Mansouri referred. All the rest of the papers “re-discovering” theTangherlini transformations were published only much later, commenc-ing in the 1990’s, so the absence of priority in those papers was notfound somewhere being discussed in the scientific literature.

Interestingly, Frank Robert Tangherlini met Stefan Marinov in per-son at the General Relativity 9th Meeting in Jena, Germany, in 1980.Tangherlini wrote, in his private letter dated October 14, 2006, abouthow this happened [101]:

“I met Marinov under a most curious circumstance: he had putup over the doorway of a hall, where many passed through, aposter of about 1/3 meter wide and about 2 meters long in whichhe criticized me, in artistic calligraphy, for not having followedup on my transformations. I found this very strange behaviour.After all, why didn’t he write directly to me, or arrange a meetingat a conference? So I suspected then he was somewhat crazy,although possibly artistically talented. With any crazy person,one shouldn’t spend too much time on him except as an exampleof how people in science, just as in every day life, can go astray”.

∗In his Nuovo Cimento article of 1961 [11], Tangherlini wrote: “Finally we shouldnote that the usual results of special relativity can obtained from the line element(1.17) and co-ordinate transformation (1.16), as we have already shown for the prob-lem of sending light signals out and back”.


In recent years, a second wave of increasing interest in the Tangher-lini transformations has risen due to the possibility of a small anisotropyof the velocity of light claimed by the Grenoble group of experimen-talists [102, 103] (see also [104–106]). At the present time, there areneither definitely verified experimental facts nor fundamental principlesof physics which could require the failure of the Lorentz-invariance ininertial reference frames (see [33, 107], for instance). Meanwhile, physi-cists are still continuing experimental and theoretical attempts to findviolations of Lorentz invariance, and also theoretical grounds to thesein the course of interpretation of bizarre physical phenomena such asthose in cosmology, quantum gravity, quantum field theory, particlephysics, space beam physics and super-high energy physics (see, in par-ticular, [18, 19, 107]). One regularly connects this possible violation,without which CPT-invariance of quantum field theory and the lawof charge conservation of classical electrodynamics cannot be violated,with a possible violation of the space-time symmetry due to, say, pro-cesses at the Planck (small) scale or due to additional (hypothetical)measurements producing a new vector or tensor field which acts ontophysical bodies depending on their velocity and orientation in space(which is different for particles and anti-particles). As a result, theoret-ical physicists expect various new effects such as a length contractionand time dilation in addition to the Lorentz ones, a variation of the elec-tromagnetic field polarization, a non-zero rest-mass of photons, changesof the masses of decaying particles and of their decay channels, and alsomany other effects which depend on the motion of the inertial frameof reference wherein the processes occur. The simplest case of theoriesthat violate Lorentz invariance is the so-called Doubly Special Theory ofRelativity (see [20,108–111], for instance), wherein elementary particlescannot be accelerated up to a velocity exceeding the velocity of light,nor can they acquire an energy exceeding a fixed numerical value spe-cific to each particle (the so-called Planck energy). The aforementionedvector or tensor field has no direct connexion to the gravitational field.Whether such possible violations of Lorentz invariance would lead tochanges of the gravitational field of a moving body, or to changes of theproperties of a black hole that are not predicted by the General Theoryof Relativity are issues for further research.

Putting aside gravitation, it is useful to study various consequencesof the violation of Lorentz invariance. In particular, as already discussedabove, the possibility of introducing alternative methods of synchroniza-tions in a given inertial frame. In this regard, it is important to keepin mind that whether clocks are synchronized according to the Einstein


procedure, or externally, one has not changed inertial frames, but ef-fectively one has merely introduced another set of clocks in the sameinertial frame. In addition to the Tangherlini synchronization which hasalready been described, still another form of synchronization has beensuggested by Torgny Sjodin in his paper of 1979 [5] in which clocks aresynchronized etc. The discussion in Chapter 6 of Tangherlini’s thesis,entitled Measurements with Signals Travelling with Finite Velocities, tosome extent anticipates Sjodin’s considerations, in that Tangherlini con-siders the possibility of other signals propagating with constant speedsgreater than or less than the speed of light relative to the rest frame.∗

Even if digressing from gravitation and other effects of the GeneralTheory of Relativity, different scenarios of the violation of the Lorentz-invariance are useful to be studied on the basis of not only the Lorentzframes of reference (their preference is due to the Einsteinian methodof synchronizations of clocks, where the out and back travel times forlight are equal), but also on the basis of other inertial frames in whichthere has been an alternative synchronization of clocks. In particular,the time dilation is to be considered/described by the use of a frameof reference whose clocks are synchronized by infinite speed signals (inpractice — a respective light spot). Such reference frames were studiedby Tangherlini, when he compared descriptions of physical processesobtained in such a reference frame to the well-known Lorentz descrip-tion. A larger class of alternatively synchronized inertial frames, whereclocks are synchronized by signals travelling with a finite speed whichcan exceed the velocity of light, was suggested later by Torgny Sjodinin his paper of 1979 [5].

The Tangherlini transformations and also the Sjodin transformationswhich generalize them gave rise to a substantial discussion a quartercentury ago, and then found respective places in the Special Theory ofRelativity. Despite the fact that the Tangherlini and Sjodin transforma-tions can yield the same results as the Special Theory of Relativity, thesetransformations are more complicated than the Lorentz transformationsince they don’t leave the speed of light invariant. However, physicistswill probably turn to these transformations each time when there is

∗In this concern, Tangherlini writes [42]: “. . . changing synchronization does not

change the inertial frame. Think of it this way. You have a train moving with con-stant velocity relative to the railroad station. You may synchronize clocks accordingto Einstein on the train, or according to my method, which is related by a local timetransformation to the Einstein synchronization, or to that of Reichenbach, or to thatof Sjodin, but that doesn’t change the uniform motion of the train. It is only whenone considers transformations from, say, the station to the train, or vice versa thatone has changed inertial frames”.


even the smallest chance that they are encountering Lorentz-invarianceviolating effects in their experiments.

The anisotropy of the coordinate velocity of light c ′ (3), measuredin a moving inertial frame of reference K ′, is the price one has to payto keep simultaneity unchanged between all inertial frames of reference.Note: within a given inertial frame, there is agreement everywhere inthat frame as to when two events are simultaneous, after the clocks haveall been synchronized, say, by the Einstein method, and in this sensesimultaneity is absolute within a given frame. It is whether simultaneitywithin one frame agrees with simultaneity within another frame that theproblem of relative simultaneity arises.

Because the Tangherlini transformations are linear, Maxwell’s equa-tions are invariant with respect to the transformations. Meanwhile, asshown by Tangherlini [4], in a moving inertial reference frame K ′ aneffective “optical medium” appears which makes the velocity of lightdifferent in the forward and backward directions, with respect to themotion of K ′. Hence, the Tangherlini transformations in common withthe Lorentz transformations can provide adequate description of phys-ical processes in a moving inertial frame of reference, but the Lorentztransformations are more useful in this deal because they keep the ve-locity of light constant and isotropic in all inertial frames of reference.

Finally, it should be mentioned that Tangherlini in his thesis usedthe fact that since Maxwell’s equations can be written in generally co-variant form, they obviously hold under his transformations as well asfor the Lorentz transformations. However, because his transformationsare linear and unimodular, as are the Lorentz and Galilean transfor-mations, and also include the Lorentz contraction and time dilation,which the Galilean transformations do not, he found that despite thedifference with the Lorentz Transformation as to synchronization, a setof tensorial expressions for the electromagnetic fields could be extractedthat were exactly the same as for the Lorentz transformation, so thatthe equations of motion of a charged particle, when written in termof proper velocity and proper acceleration (i.e., derivatives taken withrespect to proper time) could be written so as to take the same form

as for the Lorentz transformation, and, importantly do not involve thevelocity of the moving frame relative to the rest frame. He points outin the thesis that there exist a second set of equations of motion whichdo not reduce to the equations of motion, as seen by the observer inthe moving frame who uses the Lorentz transformation, that explicitlyinvolve the velocity of the moving frame relative to the rest frame, butthat in the absence of any way to synchronize the clocks in the moving


frame with the rest frame, these equations of motion are unobservables.He also points out that the d’Alembertian operator is not invariant un-der his transformation and that this is a consequence of the fact that theone-way velocity of light in the moving frame has not remained invariantin the moving frame as is the case for the Lorentz transformation, butthat in the absence of the possibility of synchronization with the restframe, this anisotropy is unobservable, in agreement with observation.

The authors’ exclusive thanks go to Frank Robert Tangherlini for theoriginal manuscript of his Dissertation of 1958 which he friendly pro-vided to us many years before it was submitted for publication, and alsofor his prior permission for translation of his Dissertation into Russian(the translation was produced by Natalia V. Roudik, with contributionof Edward G. Malykin, edited by Gregory B. Malykin; it is coming to bepublished soon, with Tangherlini’s preface and comments). We thankFrank Robert Tangherlini for the years spent on friendly discussions,and send him our best wishes. We also thank Valentin M. Gelikonovwho supported this study, Vladimir V. Kocharovski for useful notes,and also Vera I. Pozdnyakova and Natalia V. Roudik who assisted us.Special thanks go to Dmitri Rabounski for the discussion.

This work was partly supported by the Council on President’s Grantsof the Russian Federation for Leading Scientific Schools (project no.NSh. 1931.2008.2).

Submitted on June 20, 2009

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Vol. 2, 2009 ISSN 1654-9163

THE

ABRAHAM ZELMANOV

JOURNALThe journal for General Relativity,

gravitation and cosmology

TIDSKRIFTEN

ABRAHAM ZELMANOVDen tidskrift for allmanna relativitetsteorin,

gravitation och kosmologi

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